Floating booms will NOT stop oil, but they WILL redirect it to where you want. You can minimise the damage by using booms to direct the floating oil away from areas and concentrating it in catchment basins. You can see how to do this by reading Fishgrease' Fucking Boom School. Once the oil is contained you can start the separation process.
I would use banks of coalescing plate oil separators that empty to decanting drums. Kevin Costner is hawking his multi-million dollar "Ocean Therapy", but 5 minutes of reading tells me he has reinvented the Hydrocyclone. I've used these - you can buy them off the shelf - in the equipment I sell to strip oils and silt from water. They are massively complicated and very expensive, and do not give you 100% clean water. You get a better result from letting the liquid sit in a settlement pit for 24 hours and letting good old uncomplicated cheap gravity do the work.
Coalescing plate oil separator.
Decant Drum. Oil/water from separator goes into the top, water goes out the bottom, and oil stays in the drum. When the drum is full you plug the holes and install a new drum. These can be made from a standard 44 gallon (or 50 US gallon) drum and some PVC pipe.
So, after using the booms to direct the oil into collection basins, I would use banks of surface skimmers, diaphragm pumps and coalescing plate oil separators - the more of these, the more litre per minute you can treat - to separate the oil and dump it into decant drums, which would do further by-gravity separation. The drums are capped when they are full and sent to be processed at an oil refinery. All this is off-the-shelf hardware, available right now.
Oh, and those plumes of oil under the water? That's what dispersant does. That's what dispersant is supposed to do: break the oil into little drops and sink them into the water column so that the oil disperses and doesn't form a huge surface slick. The volatile fractions eventually evapourate leaving tar balls that sink or wash ashore. Did people think the dispersant made the oil magically vanish?
ETA: see what happens when I try to add content instead of just links? BORING.